In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x⁴ + 1
x³ - 2x

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To determine if a function is even, odd, or neither, we need to analyze the function's symmetry. A function y = f(x) is even if f(-x) = f(x) for all x in the domain, and it is odd if f(-x) = -f(x) for all x in the domain.

Start by substituting -x into the function y = x⁴ + 1x³ - 2x. This gives us y = (-x)⁴ + 1(-x)³ - 2(-x).

Simplify the expression: (-x)⁴ = x⁴, (-x)³ = -x³, and -2(-x) = 2x. So, the expression becomes y = x⁴ - x³ + 2x.

Compare the simplified expression y = x⁴ - x³ + 2x with the original function y = x⁴ + 1x³ - 2x. Since f(-x) ≠ f(x) and f(-x) ≠ -f(x), the function is neither even nor odd.

Conclude that the function y = x⁴ + 1x³ - 2x is neither even nor odd based on the symmetry analysis.

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Even Functions

A function is considered even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. For example, the function y = x² is even because substituting -x yields the same result as substituting x.

Odd Functions

A function is classified as odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. An example of an odd function is y = x³, as substituting -x results in the negative of the original function.

Neither Even Nor Odd Functions

A function is neither even nor odd if it does not satisfy the conditions for either classification. This means that the function's graph lacks symmetry with respect to both the y-axis and the origin. For instance, the function y = x + 1 is neither even nor odd, as it does not exhibit the required symmetries.

In Exercises 9–16, determine whether the function is even, odd, or neither.𝔂 = x⁴ + 1x³ - 2x